AbstractWe give polyhedral realizations in E3 of two regular maps with automorphism group of order 192 by W. Dyck and F.A. Sherk. The polyhedra have self-intersections as the Kepler-Poinsot solids and maximal possible symmetry group. The polyhedral realization of Dyck's map is a dodecahedron built up of 12 octogons. Its visible part is a well-known object of delusion

Wills, J.M

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A combinatorially regular dodecahedron of genus 3 

Discrete Mathematics 67 (1987) 199-204 
North-Holland 
199 
A COMBINATORIALLY REGULAR DODECAHEDRON 
OF GENUS 3 
J.M. WILLS 
Mathematics Institute, University of Siegen, D-5900 Siegen, Fed. Rep. Germany 
Received 11 October 1985 
Revised 22 October 1986 
We give polyhedral realizations in E 3 of two regular maps with automorphism group of order 
192 by W. Dyck and F.A. Sherk. The polyhedra have self-intersections as the Kepler-Poinsot 
solids and maximal possible symmetry group. The polyhedral realization of Dyck's map is a 
dodecahedron built up of 12 octogons. Its visible part is a weU-known object of delusion. 
1. Introduction 
A map is a cell-decomposition f a dosed surface into topological polygons 
called faces by means of arcs, called edges. The intersections of the edges are said 
to be the vertices of ~t. By a flag of ~ we mean as usual a triplet consisting of a 
vertex, an edge and a face, which are mutually incident. 
If the automorphism group of ~ (consisting of the incidence-preserving 
bijections of the set of vertices, edges and faces of ~)  acts transitively on the 
flags, then ~ is called regular (cf. Coxeter and Moser [3] or Garbe [5]). As for all 
regular maps (or compact Riemann surfaces) of genus g/> 2, the usual geometri- 
cal representation is in terms of their universal cover, i.e., of the Poincar6 model 
of the hyperbolic plane. 
However, from an intuitive geometrical point of view there has always been a 
particular interest in those analogues of the Platonic solids which exist as 
polyhedra in the Euclidean 3-space E 3 with or without self-intersections a e.g. 
the 4 Kepler-Poinsot solids. To be precise, we mean by a polyhedron with 
self-intersection a closed compact 2-manifold (here always oriented) in Euclidean 
3-space E 3, which is built up of finitely many plane polygons (intersecting or not). 
These polygons are bounded by finitely many line segments, which might 
intersect as e.g. a pentagram. If these polygons are such that no two adjacent 
ones are coplanar, they are called the faces of P. The vertices and edges of P are 
the vertices and edges of the faces of P, respectively. 
So the set of all vertices, edges and faces of P becomes the set of all vertices, 
edges and faces of the map on the underlying surface, In particular our definition 
rules out infinite polyhedra, branched or unbranched multiple covers of poly- 
hedra and polyhedra with non-planar faces (cf. [6]). 
0012-365X/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland) 
200 Z M. Wills 
In the following we use two notations for regular maps (cf. [1, 3, 5, 8-13]): 
{p, q}, and {p, q; g}. These means that the map consists of p-gons, q meeting at 
each vertex. Further g denotes the genus of the map and r the length of the 
Petrie-polygons, i.e., zig-zag-lines which leave alternating one face to the right 
and to the left. Although we do not need Petrie-polygons in our proofs we have 
drawn one in Fig. 1 and 4 each. 
The two polyhedral realizations of the regular maps which we will investigate, 
contain faces with self-intersections. Coxeter (cf. [2, p. 141, 142]) was perhaps 
the first authors who considered such realizations of regular maps besides the 4 
Kepler-Poinsot solids. He mentioned in [2] that two the uniform polyhedra, 
which were discovered by Hess, Pitsch and Badoureau (cf. [1, p. 115, 116]) are 
polyhedral realizations with self-intersections of the regular maps {5, 4}6 and 
{5, 6}4, both with automorphism group of order 240. If one compares polyhedral 
realizations with or without self-intersections, then the disadvantage of weaker 
topological properties (self-intersection) is compensated by higher symmetry. 
2. Dyck's regular map (8, 3}6 of genus 3 
This regular map was discovered in 1880 by W. Dyck [4]. (Although we do not 
need it in the following we mention briefly that this map occurs also in literature 
as Dyck's Riemann surface of genus 3 or as Dyck's quartic x 4 + y4 + z4 = 0.) The 
dual map {3, 8}6 has been given several times (e.g. [4], [13, Fig. 19] and [10]) and 
so it is easy to check that the map {8, 3}6 is realized by our polyhedron. Our Fig. 
1 shows Dyck's map {3, 8}6 from [10] Fig. 1. The numbers 1 , . . .  12 denote the 
vertices. As usual the edges and faces in the figure denote the edges and faces of 
the map. We now label in our polyhedron in Fig. 2 the 12 octogonal faces such 
Fig. 1. {3, 8}6. 
A combinatorially regular dodecahedron f genus 3 201 
9 
7 
/,1 
7 
~ 7 10 
Fig. 2. Dyck's {8, 3}6, f = 4(8, 12, 3). 
that they have the number of the corresponding vertex in Fig. 1. The 12 faces of 
the polyhedron split into three classes of 4 parallel equidistant faces. Faces of 
different classes are orthogonal. We label the 4 faces of each class in the following 
order: 
1, 7, 10, 4; 2, 8, 11, 5; 3, 9, 12, 6. 
In Fig. 2 we labelled the visible parts of each face. Only the faces 4, 5, 6 are 
completely invisible, whereas the only completely visible faces are 1, 2, 3. 
The faces 1 , . . . ,  6 have self-intersections and are congruent. The faces 
7 , . . . ,  12 have no self-intersections and are congruent. From Figs. 1 and 2 it is 
now easy to see that there is an incidence-reversing bijection between the 12 
vertices in Fig. 1 and the corresponding 12 octogons in Fig. 2; i.e., if and only if 
two vertices in Fig. 1 are joined by an edge, then the corresponding faces in Fig. 2 
have a common edge. From the duality follows that the polyhedron in fig. 2 is a 
realization of Dyck's {8, 3}6. 
We end this section with some remarks: 
(1) The proof of the polyhedral realization can also be given by polarity 
arguments as described in [7], and indeed this was the way we found this 
polyhedron as the polar of the polyhedron in [10] Fig. 4. 
(2) Another polyhedral realization of {8, 3}6 can be found either from the map 
or by polarity arguments from Fig. 5 in [10]. In the resulting polyhedron 
essentially the role of the inner and the outer polygons is changed. 
(3) In [10] it was shown that a polyhedral realization of {3, 8}6 with or without 
self-intersections cannot have octahedral or icosahedral symmetry. So our 
polyhedral realization has maximal symmetry group. 
(4) Perhaps the most surprising fact is that (the visible part of) our polyhedron 
is a well-known object of delusion. We demonstrate his by another projection of 
202 J.M. Wills 
Fig. 3. Dyck's {8, 3}6, f = 4(8, 12, 3). 
our octogon-dodecahedron in Fig. 3. The pairwise orthogonal axes help for the 
orientation. 
3. Sherk's regular map {6, 4; 5} 
This map is considered in [12, p. 18; 5, p. 54 and 11]. As for Dyck's map we 
start in Fig. 4 from the dual map {4, 6; 5} which was given in [11, Fig. 9]. The 
numbers 1 , . . . ,  8, 1 ' , . . . ,  8' denote the 16 vertices of {4, 6; 5}. 
We now describe briefly the polyhedron in Fig. 5. It has the full octahedral 
symmetry group. Its convex hull is one of the Archimedean solids where the 6 
square-faces are omitted. The remaining 8 faces are regular hexagons and are 
-, 7' 2' 2~ ~"~2 2 ' 5' ,., 
- 2 ,Z - 
Fig. 4. {4, 6; 5}. 
A combinatorially regular dodecahedron f genus 3 203 
Fig. 5. Sherk's {6, 4; 5}, f = 8(3, 6, 2). 
labelled 1 , . . . ,  8 (the face 2 lies on the bottom of Fig. 5). The 8 inner hexagons 
are all congruent; they have in alternating order 3 short edges which they share 
with outer hexagons, and 3 long edges which intersect each other, such that each 
of the inner hexagons forms 4 congruent regular triangles. 
The 16 faces of the polyhedron split into four classes of parallel faces, namely 
of two outer and two inner each. We label the 4 faces of each of the four classes 
in the following order: 
1, 5', 1', 5, 2, 6', 2', 6; 3, 7', 9', 7; 4, 8', 4', 8. 
In Fig. 5 we labelled the visible parts of the 16 faces. The only completely 
visible faces are 4, 5, 6, 7. The faces 1', 2', 3', 8' can be seen partially, where they 
form a pit. 
After the description of the polyhedron it is again easy to see that there is an 
incidence-reversing bijection between the 16 vertices in Fig. 4 and the cor- 
responding 16 hexagons in Fig. 5. So the polyhedron in Fig. 5 is a realization of 
Sherk's map. Again this can also be shown by polarity arguments. The 
polyhedron has complete octahedral symmetry group (of order 48), which is 
obviously the maximal possible symmetry group. 
References 
[1] H.S.M. Coxeter, Regular polytopes, 3rd ed. (Dover, New York, 1973). 
[2] H.S.M. Coxeter, The abstract groups G "'n'p, Trans. Amer. Math. Soc. 45 (1937) 74-150. 
[3] H.S.M. Coxeter and W.O.J. Moser, Generators and relations for discrete groups, 4th ed. 
(Springer, Berlin, 1980). 
[4] W. Dyck, Notiz fiber eine regul~re Riemann~he Fliche vom Geschlecht 3 und die zugeh6rige 
Normalkurve 4, Ordnung Math. Ann. 17 (1880) 510-516. 
[5] D. Garbe, ~er  die regularen Zerlegungen geschlossener orientierbarer Fliichen, J. reine und 
Angew. Math. 237 (1969) 39--55. 
[6] B. Grfinbaum, Regular polyhedra- old and new; Aequat. Math. 16 (1977) 1-20. 
204 J.M. Wills 
[7] B. Griinbaum and G. Shephard, Duality of Polyhedra; in: G. Fleck and M. Senechal, eds., 
Shaping Space - A polyhedral approach (Birkh[iuser, Basel) in print. 
[8] E. Schulte and J.M. Wills, A polyhedral realization of Felix Klein's map {3, 7}8 on a Riemann 
surface of genus 3, J. London Math. Soc. (2) 32 (1985) 539-547. 
[9] E. Schulte and J.M. Wills, On Coxeter's regular skew polyhedra, Discrete Math. 60 (1986) 
253-262. 
[10] E. Schulte and J.M. Wills, Geometric realizations for Dyck's regular map on a surface of genus 
3, J. Discrete Comput. Geom. 1 (1986) 141-153. 
[11] E. Schulte and J.M. Wills, Kepler-Poinsot-type realizations of regular maps of Klein, Gordan 
and Sherk, to appear. 
[12] F.A. Sherk, A family of regular maps of type {6, 6}, Canad. Math. Bull. 5 (1962) 13-20. 
[13] F.A. Sherk, The regular maps on a surface of genus three, Canad. J. Math. 11 (1959) 452-480. 


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A combinatorially regular dodecahedron of genus 3

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